The First Law of Thermodynamics

What is the first law of thermodynamics?

Many power plants and engines operate by turning heat energy into work. The reason is that a heated gas can do work on mechanical turbines or pistons, causing them to move. The first law of thermodynamics applies the conservation of energy principle to systems where heat transfer and doing work are the methods of transferring energy into and out of the system. The first law of thermodynamics states that the change in internal energy of a system $\Delta U$ equals the net heat transfer into the system $Q$, plus the net work done on the system $W$. In equation form, the first law of thermodynamics is,

$\Large \Delta U=Q+W$

[Wait, why did my book/professor use a negative sign in this equation?]

$\Delta U=Q+W$$\Delta U=Q-W$

$\Delta U=Q+W_\text{on gas}$$W$$\Delta U=Q-W_\text{by gas}$$W$

$W_\text{on gas}=-W_\text{by gas}$

$\Delta U=Q+W_\text{on gas}$$\Delta U=Q-W_\text{by gas}$

$\Delta U=Q+W_\text{on gas}$$W$

Here $\Delta U$ is the change in internal energy $U$ of the system. $Q$ is the net heat transferred into the system—that is, $Q$ is the sum of all heat transfer into and out of the system. $W$ is the net work done on the system.

So positive heat $Q$ adds energy to the system and positive work $W$ adds energy to the system. This is why the first law takes the form it does, $\Delta U=Q+W$. It simply says that you can add to the internal energy by heating a system, or doing work on the system.

What do each of these terms ($\Delta U, Q, W$) mean?

Nothing quite exemplifies the first law of thermodynamics as well as a gas (like air or helium) trapped in a container with a tightly fitting movable piston (as seen below). We'll assume the piston can move up and down, compressing the gas or allowing the gas to expand (but no gas is allowed to escape the container).

The gas molecules trapped in the container are the "system". Those gas molecules have kinetic energy. 

[Can a gas molecule have potential energy?]

$O_2$

$U$

The internal energy $U$ of our system can be thought of as the sum of all the kinetic energies of the individual gas molecules. So, if the temperature $T$ of the gas increases, the gas molecules speed up and the internal energy $U$ of the gas increases (which means $\Delta U$ is positive). Similarly, if the temperature $T$ of the gas decreases, the gas molecules slow down, and the internal energy $U$ of the gas decreases (which means $\Delta U$ is negative).

It's really important to remember that internal energy $U$ and temperature $T$will both increase when the speeds of the gas molecules increase, since they are really just two ways of measuring the same thing; how much energy is in a system. Since temperature and internal energy are proportional $T \propto U$, if the internal energy doubles the temperature doubles. Similarly, if the temperature does not change, the internal energy does not change.

One way we can increase the internal energy $U$ (and therefore the temperature) of the gas is by transferring heat $Q$ into the gas. We can do this by placing the container over a Bunsen burner or submerging it in boiling water. The high temperature environment would then conduct heat thermally through the walls of the container and into the gas, causing the gas molecules to move faster. If heat enters the gas, $Q$ will be a positive number. Conversely, we can decrease the internal energy of the gas by transferring heat out of the gas. We could do this by placing the container in an ice bath. If heat exits the gas, $Q$ will be a negative number. This sign convention for heat $Q$ is represented in the image below.

Since the piston can move, the piston can do work on the gas by moving downward and compressing the gas. The collision of the downward moving piston with the gas molecules causes the gas molecules to move faster, increasing the total internal energy. If the gas is compressed, the work done on the gas $W_\text{on gas}$ is a positive number. Conversely, if the gas expands and pushes the piston upward, work is done by the gas. The collision of the gas molecules with the receding piston causes the gas molecules to slow down, decreasing the internal energy of the gas. If the gas expands, the work done on the gas $W_\text{on gas}$ is a negative number. This sign convention for work $W$ is represented in the image below.

Below is a table that summarizes the signs conventions for all three quantities $(\Delta U, Q, W)$ discussed above.

$\Delta U$ (change in internal energy)	$Q$ (heat)	$W$ (work done on gas)
is $+$ if temperature $T$increases	is $+$ if heat enters gas	is $+$ if gas is compressed
is $-$ if temperature $T$decreases	is $-$ if heat exits gas	is $-$ if gas expands
is $0$ if temperature $T$ is constant	is $0$ if no heat exchanged	is $0$ if volume is constant

Is heat $Q$ the same thing as temperature $T?$

Absolutely not. This is one of the most common misconceptions when dealing with the first law of thermodynamics. The heat $Q$ represents the heat energy that enters a gas (e.g. thermal conduction through the walls of the container). The temperature $T$ on the other hand, is a number that's proportional to the total internal energy of the gas. So, $Q$ is the energy a gas gains through thermal conduction, but $T$ is proportional to the total amount of energy a gas has at a given moment. The heat that enters a gas might be zero $(Q=0)$ if the container is thermally insulated, however, that does not mean that the temperature of the gas is zero (since the gas likely had some internal energy to start with).

To drive this point home, consider the fact that the temperature $T$ of a gas can increase even if heat $Q$ leaves the gas. This sounds counterintuitive, but since both work and heat can change the internal energy of a gas, they can both affect the temperature of a gas. For instance, if you place a piston in a sink of ice water, heat will conduct energy out the gas. However, if we compress the piston so that the work done on the gas is greater than the heat energy that leaves the gas, the total internal energy of the gas will increase.

What do solved examples involving the first law of thermodynamics look like?

Example 1: Nitrogen piston

A container has a sample of nitrogen gas and a tightly fitting movable piston that does not allow any of the gas to escape. During a thermodynamics process, $200\text{ joules}$ of heat enter the gas, and the gas does $300 \text{ joules}$ of work in the process.

What was the change in internal energy of the gas during the process described above?

Solution:
We'll start with the first law of thermodynamics.

$\Delta U=Q+W \quad \text{(start with the first law of thermodynamics)}$

$\Delta U=(+200 \text{ J})+W \quad \text{(plug in }Q=+200\text{ J)}$ 

[Why is heat a positive number here?]

$Q$

$\Delta U=(+200 \text{ J})+(-300\text{ J}) \quad \text{(plug in }W=-300\text{ J)}$

[Why is work a negative number here?]

$\Delta U=-100\text{ J} \quad \text{(calculate and celebrate)}$

Note: Since the internal energy of the gas decreases, the temperature must decrease as well.

Example 2: Heating helium

Four identical containers have equal amounts of helium gas that all start at the same initial temperature. Containers of gas also have a tightly fitting movable piston that does not allow any of the gas to escape. Each sample of gas is taken through a different process as described below:

Sample 1: $500 \text{ J}$ of heat exits the gas and the gas does $300 \text{ J}$ of work
Sample 2: $500 \text{ J}$ of heat enters the gas and the gas does $300 \text{ J}$ of work
Sample 3: $500 \text{ J}$ of heat exits the gas and $300 \text{ J}$ of work is done on the gas
Sample 4: $500 \text{ J}$ of heat enters the gas and $300 \text{ J}$ of work is done on the gas

Which of the following correctly ranks the final temperatures of the samples of gas after they're taken through the processes described above.

A. $T_4>T_3>T_2>T_1$

B. $T_1>T_3>T_2>T_4$

C. $T_4>T_2>T_3>T_1$

D. $T_1>T_4>T_3>T_2$

Solution:
Whichever gas has the largest increase in internal energy $\Delta U$ will also have the greatest increase in temperature $\Delta T$ (since temperature and internal energy are proportional). To determine how the internal energy changes, we'll use the first law of thermodynamics for each process.

Process 1:
$\begin{aligned} \Delta U&=Q+W \\ \Delta U&=(-500\text{ J})+(-300\text{ J}) \\ \Delta U&=-800\text{ J} \end{aligned}$

Process 2:
$\begin{aligned} \Delta U&=Q+W \\ \Delta U&=(+500\text{ J})+(-300\text{ J}) \\ \Delta U&=+200\text{ J} \end{aligned}$

Process 3:
$\begin{aligned} \Delta U&=Q+W \\ \Delta U&=(-500\text{ J})+(300\text{ J}) \\ \Delta U&=-200\text{ J} \end{aligned}$

Process 4:
$\begin{aligned} \Delta U&=Q+W \\ \Delta U&=(+500\text{ J})+(+300\text{ J}) \\ \Delta U&=+800\text{ J} \end{aligned}$

The final temperatures of the gas will have the same ranking as the changes in internal energy (i.e. sample 4 has the largest increase in internal energy, so sample 4 will end with the largest temperature).

$\Delta U_4>\Delta U_2>\Delta U_3>\Delta U_1$ and $T_4>T_2>T_3>T_1$

So the correct answer is C.